Description
Two-dimensional unitary chiral conformal field theories (CFTs) admit three distinct mathematical formulations: unitary vertex operator algebras (uVOAs), conformal nets, and Segal (functorial) chiral CFTs. With the aim of building a fully extended functorial chiral CFT from the data of a conformal net, we give its values on points and 1-dimensional cobordisms: to a point, we assign the category of solitonic representations of the net, and to a 1-dimensional cobordism, a bimodule category. We then prove that this assignment is functorial, with gluing of cobordisms corresponding to fusion of modules. The algebraic target of the construction is the 3-category BicomCat of bicommutant categories, the functional-analytic analogue of TensCat. We introduce the fusion of modules over bicommutant categories as 'categorified' Connes fusion.